REVIEW
On Greenberg's conjecture for rational elliptic curves at Eisenstein primes
Number Theory · Submitted 30 Sep 2026 · 33 references
Pith's verdict: The algebraic part looks like a genuine advance — case (ii) of Greenberg's conjecture at arbitrary odd ordinary Eisenstein primes — but the analytic half leans on an unpublished preprint [KY24] whose stated scope (anticyclotomic, semistable) doesn't obviously cover the cyclotomic good-ordinary branch it is asked to carry.
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Referee report
The paper
The paper claims
For rational elliptic curves at odd good ordinary Eisenstein primes, the paper proves Greenberg's conjecture: an isogenous curve has vanishing algebraic Iwasawa μ-invariant, and via the Iwasawa Main Conjecture the analytic μ-invariant…
The reading
What carries the argument
Iwasawa $\mu$-invariants count the $p$-power divisibility of the characteristic power series of a finitely generated torsion module over the Iwasawa algebra $\Lambda=\mathbb{Z}_p[\![\,\Gamma\,]\!]$; for a Selmer group, $\mu=0$ means the $p$-primary Tate–Shafarevich groups stay bounded along the cyclotomic tower, i.e., $\#\Sha(E(\mathbb{Q}_n))[p]$ is bounded as $n\to\infty$. The load-bearing mechanism is a commutative diagram of global-to-local maps that tracks the ' $\mu=1$' part of $H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_\infty,E[p])$ through restriction to $L=\mathbb{Q}(E[p])$, where $E[p]$ splits as $F(\varphi)\oplus F(\psi)$, and then through local evaluation at primes above $p$. Two finiteness results close the argument: $\ker(\mathrm{proj}_L\circ \mathrm{res}_{L/\mathbb{Q}})$ is finite because the non-splitness of the extension makes $F(\psi)$ fail to be $\mathrm{Gal}(L/\mathbb{Q})$-invariant, and $\ker(\mathrm{loc}_{L,F(\varphi)})$ is finite by the vanishing of the fine Selmer $\mu$ over $L$. The p-extension principle (Proposition 3.1.5), proved via Nakayama's lemma, is what transfers $\mu=0$ from the abelian field $\mathbb{Q}(\varphi,\psi)$ — where Ferrero–Washington applies — up to the p-extension $L$.
Extended reading notes
Core claim
Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd good ordinary Eisenstein prime, meaning $E[p]$ fits into a short exact sequence $0\to F(\varphi)\to E[p]\to F(\psi)\to 0$ of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-modules with $\varphi\psi=\omega$. Of the two unresolved Eisenstein cases, case (i) — one of $\varphi,\psi$ unramified and odd — was settled by Greenberg–Vatsal [GV00]; this paper settles case (ii), where $\varphi$ is unramified and even, $\psi$ ramified and odd, and the extension is non-split. Over the intermediate abelian field $\mathbb{Q}(\varphi,\psi)$ the classical, classical-fine, and fine-for-$E$ $\mu$-invariants all vanish by the Ferrero–Washington theorem, and a Nakayama-based p-extension principle (Proposition 3.1.5) lifts this vanishing up to $L=\mathbb{Q}(E[p])$, where $E[p]$ splits as $F(\varphi)\oplus F(\psi)$. With these vanishings in hand, the paper tracks where the ' $\mu=1$' part of the global-to-local image of $H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_\infty,E[p])$ lands: the non-splitness of the extension makes $F(\psi)$ fail to be $\mathrm{Gal}(L/\mathbb{Q})$-invariant, so restriction kills the intersection with the ramified summand, and the fine-$\mu$ vanishing kills the obstruction locally, together forcing Greenberg's $\mu$ over $\mathbb{Q}$ to vanish. Composing with the Iwasawa Main Conjecture at Eisenstein primes ([GV00], [CGS25], [KY24]) yields the analytic vanishing (Theorem 3.3.1) and confirms Stevens's conjecture. Theorem B records the vanishing of classical, classical-fine, and fine-for-$E$ $\mu$-invariants over $\mathbb{Q}(E[p])$, and Theorem C extends the reduction principle to the residually irreducible case, where vanishing of the classical fine $\mu$ over $\mathbb{Q}(E[p])$ would force Greenberg's $\mu$ over $\mathbb{Q}$ to vanish.
Load-bearing premise
The load-bearing step is the reduction chain in section 3.1.3: if the fine Selmer $\mu$-invariant over $L=\mathbb{Q}(E[p])$ vanishes, then Greenberg's $\mu$ over $\mathbb{Q}$ vanishes, which rests on the identification $\mathrm{Gal}(L_\infty/\mathbb{Q}_\infty)\cong\mathrm{Gal}(L/\mathbb{Q})$ (the claim that $L\cap\mathbb{Q}_\infty=\mathbb{Q}$, justified by a field degree comparison plus a contradiction argument) together with the Nakayama p-extension principle and the assertion that the ' $\mu=1$' part of $H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_\infty,E[p])$ does not land entirely in the ramified summand $F(\psi)$; if any of these premises fails, the tracking mechanism collapses.
Editorial extensions
If this is right
- At every odd good ordinary Eisenstein prime, some curve in the isogeny class has vanishing algebraic $\mu$-invariant (Theorem A), and — via the Iwasawa Main Conjecture ([GV00], [CGS25], [KY24]) — vanishing analytic $\mu$-invariant for the Mazur–Swinnerton-Dyer $p$-adic $L$-function (Theorem 3.3.1), confirming Stevens's conjecture.
- Over the p-division field $L=\mathbb{Q}(E[p])$, the classical, classical-fine, and fine-for-$E$ $\mu$-invariants all vanish (Theorem B), so the classical Selmer structure over $L$ carries no $p$-divisibility at all.
- In the multiplicative reduction case, the algebraic part of Greenberg's conjecture holds already; the analytic part follows once the Iwasawa Main Conjecture is extended to multiplicative primes, as the authors note.
- Because algebraic and analytic $\mu$ agree at Eisenstein primes (Corollary 3.2.2), all isogeny-class behaviour of $\mu$ is captured by Schneider's formula: positivity arises only from ramified-odd submodules of $E[p^\infty]$, giving the bound $\mu_E\le n$ when $E[p^n]$ is reducible and $E[p^{n+1}]$ is irreducible.
- Theorem C gives a general reduction in the residually irreducible case: Greenberg's $\mu$ over $\mathbb{Q}$ vanishes as soon as the classical fine $\mu$ over $\mathbb{Q}(E[p])$ vanishes, so the outstanding problem becomes a pure number-field statement.
Reading between the lines
- If the reduction stands, the Eisenstein case of Greenberg's conjecture is effectively reduced to a classical number-field finiteness statement (Ferrero–Washington over an abelian field plus a p-extension lift), which suggests that further work — e.g., explicit bounds on $\lambda$ or uniformity across isogeny classes — could proceed through classical Iwasawa theory rather than Selmer-group structur
- The one place I would scrutinize is the degree-comparison argument for $L\cap\mathbb{Q}_\infty=\mathbb{Q}$ in Step I: in the generic case $\mathbb{Q}(E[p])\neq\mathbb{Q}(\varphi,\psi)$ it rests on a contradiction between abelian and non-abelian fields, and a hands-on computation of $[\mathbb{Q}(E[p])\cap\mathbb{Q}_\infty:\mathbb{Q}]$ on a concrete curve would settle it.
- Theorem C effectively reframes the residually irreducible case as a conjecture about classical fine $\mu$-invariants over division fields — a statement about class groups and cyclotomic towers rather than Selmer groups — which might be easier to attack computationally.
- The tracking diagram uses almost nothing specific to elliptic curves beyond the shape of $E[p]$, so the same mechanism should extend to higher-weight modular forms with reducible residual Galois representation, with the local splitting and fine-Selmer inputs as the main obstacles; the authors flag this as later work.
Original abstract
Let $E/\mathbb{Q}$ be an elliptic curve, and let $p$ be an odd prime of ordinary reduction for $E$, and assume that $E$ admits a rational $p$-isogeny. In this paper we prove Greenberg's conjecture on the vanishing of algebraic Iwasawa $\mu$-invariants of the Selmer groups attached to $E$ over the cyclotomic $\mathbb{Z}_p$-extension of the rational numbers. Via the Iwasawa Main Conjecture for elliptic curves at Eisenstein primes, we also confirm Stevens's conjecture on the behavior of analytic $\mu$-invariants.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Let $E/\mathbb{Q}$ be an elliptic curve, and let $p$ be an odd prime of ordinary reduction for $E$, and assume that $E$ admits a rational $p$-isogeny. In this paper we prove Greenberg's conjecture on the vanishing of algebraic Iwasawa $\mu$-invariants of the Selmer groups attached to $E$ over the cyclotomic $\mathbb{Z}_p$-extension of the rational numbers. Via the Iwasawa Main Conjecture for elliptic curves at Eisenstein primes, we also confirm Stevens's conjecture on the behavior of analytic $\mu$-invariants.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $E/\mathbb{Q}$ be an elliptic curve, and let $p$ be an odd prime of ordinary reduction for $E$, and assume that $E$ admits a rational $p$-isogeny. | background 1.00 | 0.06 | 0.02 | asserted 0.84 |
| s1 | In this paper we prove Greenberg's conjecture on the vanishing of algebraic Iwasawa $\mu$-invariants of the Selmer groups attached to $E$ over the cyclotomic $\mathbb{Z}_p$-extension of the rational numbers. | result 0.58 | 0.96 | 0.95 | asserted 1.00 |
| s2 | Via the Iwasawa Main Conjecture for elliptic curves at Eisenstein primes, we also confirm Stevens's conjecture on the behavior of analytic $\mu$-invariants. | result 0.91 | 0.89 | 0.12 | asserted 1.00 |
What was examined, and what was not
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References
Discussion
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